Properties

Label 27.7.f.a
Level 27
Weight 7
Character orbit 27.f
Analytic conductor 6.211
Analytic rank 0
Dimension 102
CM No

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 27.f (of order \(18\) and degree \(6\))

Newform invariants

Self dual: No
Analytic conductor: \(6.21146025774\)
Analytic rank: \(0\)
Dimension: \(102\)
Relative dimension: \(17\) over \(\Q(\zeta_{18})\)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \(102q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut +\mathstrut 210q^{5} \) \(\mathstrut -\mathstrut 342q^{6} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut 1242q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(102q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut +\mathstrut 210q^{5} \) \(\mathstrut -\mathstrut 342q^{6} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut 1242q^{9} \) \(\mathstrut -\mathstrut 3q^{10} \) \(\mathstrut -\mathstrut 492q^{11} \) \(\mathstrut +\mathstrut 3549q^{12} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut -\mathstrut 12183q^{14} \) \(\mathstrut -\mathstrut 7974q^{15} \) \(\mathstrut -\mathstrut 198q^{16} \) \(\mathstrut -\mathstrut 9q^{17} \) \(\mathstrut +\mathstrut 40995q^{18} \) \(\mathstrut -\mathstrut 3q^{19} \) \(\mathstrut -\mathstrut 19767q^{20} \) \(\mathstrut -\mathstrut 2406q^{21} \) \(\mathstrut -\mathstrut 10806q^{22} \) \(\mathstrut +\mathstrut 14466q^{23} \) \(\mathstrut +\mathstrut 35352q^{24} \) \(\mathstrut -\mathstrut 15990q^{25} \) \(\mathstrut +\mathstrut 64548q^{27} \) \(\mathstrut -\mathstrut 12q^{28} \) \(\mathstrut -\mathstrut 45690q^{29} \) \(\mathstrut -\mathstrut 186003q^{30} \) \(\mathstrut +\mathstrut 18354q^{31} \) \(\mathstrut -\mathstrut 140544q^{32} \) \(\mathstrut -\mathstrut 78804q^{33} \) \(\mathstrut -\mathstrut 82278q^{34} \) \(\mathstrut +\mathstrut 268263q^{35} \) \(\mathstrut +\mathstrut 412632q^{36} \) \(\mathstrut -\mathstrut 3q^{37} \) \(\mathstrut +\mathstrut 28428q^{38} \) \(\mathstrut +\mathstrut 126402q^{39} \) \(\mathstrut -\mathstrut 34593q^{40} \) \(\mathstrut +\mathstrut 234948q^{41} \) \(\mathstrut -\mathstrut 466074q^{42} \) \(\mathstrut +\mathstrut 148764q^{43} \) \(\mathstrut -\mathstrut 1222785q^{44} \) \(\mathstrut -\mathstrut 646578q^{45} \) \(\mathstrut -\mathstrut 3q^{46} \) \(\mathstrut +\mathstrut 517098q^{47} \) \(\mathstrut +\mathstrut 1608681q^{48} \) \(\mathstrut -\mathstrut 311262q^{49} \) \(\mathstrut +\mathstrut 1710663q^{50} \) \(\mathstrut +\mathstrut 825291q^{51} \) \(\mathstrut -\mathstrut 213057q^{52} \) \(\mathstrut -\mathstrut 1089018q^{54} \) \(\mathstrut -\mathstrut 12q^{55} \) \(\mathstrut -\mathstrut 3167925q^{56} \) \(\mathstrut -\mathstrut 1657317q^{57} \) \(\mathstrut +\mathstrut 663789q^{58} \) \(\mathstrut -\mathstrut 230802q^{59} \) \(\mathstrut -\mathstrut 168786q^{60} \) \(\mathstrut -\mathstrut 51414q^{61} \) \(\mathstrut +\mathstrut 1777536q^{62} \) \(\mathstrut +\mathstrut 1460250q^{63} \) \(\mathstrut +\mathstrut 983037q^{64} \) \(\mathstrut +\mathstrut 2264754q^{65} \) \(\mathstrut +\mathstrut 1054197q^{66} \) \(\mathstrut -\mathstrut 845052q^{67} \) \(\mathstrut -\mathstrut 3979053q^{68} \) \(\mathstrut -\mathstrut 3558150q^{69} \) \(\mathstrut -\mathstrut 835251q^{70} \) \(\mathstrut -\mathstrut 427689q^{71} \) \(\mathstrut -\mathstrut 419292q^{72} \) \(\mathstrut -\mathstrut 257043q^{73} \) \(\mathstrut +\mathstrut 2458185q^{74} \) \(\mathstrut +\mathstrut 3429390q^{75} \) \(\mathstrut +\mathstrut 2759802q^{76} \) \(\mathstrut +\mathstrut 5459610q^{77} \) \(\mathstrut +\mathstrut 5939316q^{78} \) \(\mathstrut +\mathstrut 662682q^{79} \) \(\mathstrut -\mathstrut 3002958q^{81} \) \(\mathstrut -\mathstrut 12q^{82} \) \(\mathstrut -\mathstrut 4926426q^{83} \) \(\mathstrut -\mathstrut 10132716q^{84} \) \(\mathstrut -\mathstrut 2530881q^{85} \) \(\mathstrut -\mathstrut 6700758q^{86} \) \(\mathstrut -\mathstrut 3444642q^{87} \) \(\mathstrut -\mathstrut 2860230q^{88} \) \(\mathstrut +\mathstrut 5907321q^{89} \) \(\mathstrut +\mathstrut 5189148q^{90} \) \(\mathstrut -\mathstrut 451443q^{91} \) \(\mathstrut +\mathstrut 712473q^{92} \) \(\mathstrut -\mathstrut 2952102q^{93} \) \(\mathstrut +\mathstrut 7395807q^{94} \) \(\mathstrut -\mathstrut 2179119q^{95} \) \(\mathstrut +\mathstrut 809838q^{96} \) \(\mathstrut +\mathstrut 272208q^{97} \) \(\mathstrut +\mathstrut 1179270q^{98} \) \(\mathstrut +\mathstrut 1743822q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −15.2056 2.68115i −15.0389 22.4239i 163.880 + 59.6475i 37.7314 + 44.9665i 168.554 + 381.289i 398.264 144.956i −1476.19 852.278i −276.661 + 674.463i −453.165 784.905i
2.2 −12.9309 2.28007i 19.5181 + 18.6559i 101.870 + 37.0775i −78.6006 93.6725i −209.851 285.740i 149.932 54.5707i −504.967 291.543i 32.9160 + 728.257i 802.798 + 1390.49i
2.3 −11.6551 2.05510i −14.9225 + 22.5015i 71.4770 + 26.0155i 105.310 + 125.503i 220.166 231.590i −278.603 + 101.403i −123.650 71.3896i −283.637 671.559i −969.470 1679.17i
2.4 −9.97059 1.75808i 24.1318 12.1101i 36.1814 + 13.1690i 60.9980 + 72.6946i −261.899 + 78.3191i −133.791 + 48.6960i 223.553 + 129.069i 435.691 584.478i −480.383 832.047i
2.5 −9.42516 1.66191i −26.9402 1.79573i 25.9314 + 9.43824i −132.788 158.250i 250.932 + 61.6973i −462.855 + 168.465i 301.732 + 174.205i 722.551 + 96.7550i 988.548 + 1712.22i
2.6 −6.34855 1.11942i 2.13616 26.9154i −21.0893 7.67587i −34.3281 40.9106i −43.6912 + 168.482i 90.1091 32.7970i 482.595 + 278.626i −719.874 114.991i 172.137 + 298.151i
2.7 −3.87880 0.683936i −3.37914 + 26.7877i −45.5630 16.5836i −43.1394 51.4115i 31.4281 101.593i 424.254 154.416i 383.689 + 221.523i −706.163 181.039i 132.167 + 228.919i
2.8 −2.18396 0.385091i −26.6089 4.57871i −55.5189 20.2072i 71.1763 + 84.8246i 56.3496 + 20.2466i 318.690 115.994i 236.384 + 136.476i 687.071 + 243.669i −122.781 212.663i
2.9 −0.698460 0.123157i 20.3194 + 17.7798i −59.6676 21.7172i 95.6034 + 113.936i −12.0026 14.9210i −285.640 + 103.964i 78.3107 + 45.2127i 96.7599 + 722.550i −52.7432 91.3539i
2.10 1.51454 + 0.267054i 26.8676 2.67034i −57.9178 21.0804i −139.142 165.822i 41.4052 + 3.13078i −39.0374 + 14.2084i −167.329 96.6072i 714.739 143.491i −166.452 288.303i
2.11 4.56069 + 0.804173i −8.77856 25.5331i −39.9871 14.5541i 33.6911 + 40.1515i −19.5033 123.508i −487.528 + 177.446i −427.344 246.727i −574.874 + 448.287i 121.366 + 210.212i
2.12 6.39689 + 1.12795i −18.1397 + 19.9988i −20.4923 7.45860i −37.9291 45.2022i −138.595 + 107.470i −230.668 + 83.9562i −482.696 278.685i −70.9056 725.544i −191.643 331.936i
2.13 7.40620 + 1.30591i 20.7548 17.2696i −6.99392 2.54558i 93.4519 + 111.372i 176.267 100.798i 468.870 170.655i −465.300 268.641i 132.523 716.853i 546.682 + 946.881i
2.14 10.4695 + 1.84606i −18.3558 19.8006i 46.0625 + 16.7654i −137.909 164.354i −155.623 241.189i 506.532 184.363i −137.929 79.6334i −55.1285 + 726.913i −1140.44 1975.29i
2.15 11.6192 + 2.04878i 14.4502 + 22.8077i 70.6679 + 25.7210i 2.84045 + 3.38512i 121.171 + 294.613i 129.733 47.2189i 114.471 + 66.0899i −311.386 + 659.151i 26.0684 + 45.1518i
2.16 13.7878 + 2.43117i −26.9804 + 1.02752i 124.053 + 45.1517i 118.006 + 140.634i −374.500 51.4266i −10.3283 + 3.75918i 824.665 + 476.120i 726.888 55.4459i 1285.14 + 2225.93i
2.17 14.6025 + 2.57482i 19.5272 18.6464i 146.463 + 53.3083i −47.4564 56.5563i 333.158 222.005i −559.698 + 203.713i 1179.64 + 681.064i 33.6255 728.224i −547.361 948.057i
5.1 −9.07449 10.8146i −10.8701 24.7152i −23.4948 + 133.245i 72.7572 + 199.899i −168.644 + 341.833i −62.5083 354.502i 871.728 503.293i −492.683 + 537.312i 1501.58 2600.82i
5.2 −8.78575 10.4705i 21.2323 16.6790i −21.3274 + 120.954i −83.1319 228.403i −361.178 75.7744i 3.85901 + 21.8856i 696.250 401.980i 172.622 708.267i −1661.11 + 2877.12i
5.3 −8.36754 9.97205i −20.7807 + 17.2384i −18.3125 + 103.855i −10.7822 29.6238i 345.786 + 62.9829i 45.8281 + 259.904i 467.375 269.839i 134.674 716.452i −205.189 + 355.398i
See next 80 embeddings (of 102 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.17
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Hecke kernels

There are no other newforms in \(S_{7}^{\mathrm{new}}(27, [\chi])\).